US2021124320A1PendingUtilityA1
System for Continuous-Time Optimization with Pre-Defined Finite-Time Convergence
Assignee: MITSUBISHI ELECTRIC RES LABORATORIES INCPriority: Oct 28, 2019Filed: Oct 28, 2019Published: Apr 29, 2021
Est. expiryOct 28, 2039(~13.2 yrs left)· nominal 20-yr term from priority
G05B 13/041G06F 17/13G05B 17/02F24F 11/62G05B 2219/2614B25J 9/1633
39
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Claims
Abstract
A controller for controlling a system is provided. The controller performs measuring variables via an interface to generate a vector of variables, providing a cost function, with respect to the system, based on the vector variables using weighting factors, wherein the vector variables are represented by a time-step, computing first-derivative of the cost function at an initial time-step, obtaining a convergence time from the first-derivative of the cost function, computing second derivative of the cost function and generating an optimization differential equation based on the first and second derivatives of the cost function, proceeding, starting with the initial time-step, to obtain a value of the optimization differential equation by solving the optimization differential equation, in an iteration manner, with a predetermined time step being multiplied with the value of the solved differential equation to obtain next vector variables corresponding to a next iteration time-step, until the time-step reaches the convergence time, and outputting optimal values of the vector of variables and the cost function.
Claims
exact text as granted — not AI-modifiedWe claim:
1 . A controller for controlling a system comprising:
an interface configured to receive measurement signals from sensor units and output control signals to the system to be controlled; a memory to store computer-executable algorithms including variable measuring algorithm, cost function equations, ordinary differential equation (ODE) and ordinary differential inclusion (ODI) solving algorithms and Optimal variables' values output algorithm; a processor, in connection with the memory, configured to perform steps of receiving measuring variables via the interface to generate a vector of variables; providing a cost function equation, with respect to the system, based on the vector variables using weighting factors, wherein the vector variables are represented by a time-step; computing first-derivative of the cost function at an initial time-step;
obtaining a convergence time from the first-derivative of the cost function;
computing second derivative of the cost function and generating an optimization differential equation based on the first and second derivatives of the cost function;
proceeding, starting with the initial time-step, to obtain a value of the optimization differential equation or differential inclusion by solving the optimization differential equation or the differential inclusion, in an iteration manner, with a predetermined time step being multiplied with the value of the solved differential equation to obtain next vector variables corresponding to a next iteration time-step, until the time-step reaches the convergence time; and
outputting optimal values of the vector of variables and the cost function.
2 . The controller of claim 1 , wherein the optimization differential equation is solved by a first order Euler steps:
x ( k+ 1)= x ( k )+ h.F ( k,x ( k )),
where h>0 is the discretization time-step, and k=0, 1, 2, . . . , is the discretization index, here, F is the optimization differential equation or differential inclusion.
3 . The controller of claim 1 , wherein the optimization differential equation is solved by the Runge-Kutta discretization steps.
4 . The controller of claim 1 , wherein the optimization differential equation is solved by a disctization steps:
5 . The controller of claim 1 , wherein the optimization differential equation is
x
.
=
-
c
∇
f
(
x
)
p
[
∇
2
f
(
x
)
]
r
∇
f
(
x
)
∇
f
(
x
)
T
[
∇
2
f
(
x
)
]
r
+
1
∇
f
(
x
)
,
where the constant coefficient c,p,r, are such that c>0, p∈[1,2), and r∈R, and ƒ represents the cost function, ∇ƒ(x) represents the gradient of the cost function, and ∇ 2 ƒ(x) the Hessian of the cost function.
6 . The controller of claim 1 , wherein the optimization differential equation is
x
.
=
-
c
∇
f
(
x
)
1
p
-
1
[
∇
2
f
(
x
)
]
r
sign
(
∇
f
(
x
)
)
sign
(
∇
f
(
x
)
)
T
[
∇
2
f
(
x
)
]
r
+
1
sign
(
∇
f
(
x
)
)
.
Where the constant coefficient c,p,r, are such that c>0, p∈[1,2), and r∈R, and ƒ represents the cost function, ∇ƒ(x) represents the gradient of the cost function, ∇ 2 ƒ(x) the Hessian of the cost function, and sign(.) is the sign function.
7 . The controller of claim 1 , wherein the optimization differential equation is
x
.
=
-
w
(
t
,
x
;
x
0
)
[
∇
2
f
(
t
,
x
)
]
r
∇
f
(
t
,
x
)
∇
f
(
t
,
x
)
⊤
[
∇
2
f
(
t
,
x
)
]
r
+
1
∇
f
(
t
,
x
)
,
w
(
t
,
x
;
x
0
)
=
Δ
∇
f
(
0
,
x
0
)
2
(
1
-
α
)
2
(
1
-
α
)
T
∇
f
(
t
,
x
)
2
α
+
∇
f
(
t
,
x
)
⊤
∂
∇
f
∂
t
(
t
,
x
)
.
8 . The controller of claim 1 , wherein the optimization differential equation is
x
.
=
-
1
2
[
∇
2
f
(
t
,
x
)
]
r
∇
f
(
t
,
x
)
∇
f
(
t
,
x
)
T
[
∇
2
f
(
t
,
x
)
]
r
+
1
∇
f
(
t
,
x
)
(
2
l
(
t
,
x
)
∇
f
(
t
,
x
)
+
c
∇
f
(
t
,
x
)
2
α
)
,
.
9 . The controller of claim 1 , wherein the gradient and Hessian are computed using numerical differentiation.
10 . The controller of claim 1 , wherein the gradient and Hessian are computed using filters.
11 . A computer-implemented method for controlling a system comprising:
measuring variables via an interface to generate a vector of variables; providing a cost function, with respect to the system, based on the vector variables using weighting factors, wherein the vector variables are represented by a time-step; computing first-derivative of the cost function at an initial time-step;
obtaining a convergence time from the first-derivative of the cost function;
computing second derivative of the cost function and generating an optimization differential equation based on the first and second derivatives of the cost function;
proceeding, starting with the initial time-step, to obtain a value of the optimization differential equation or differential inclusion by solving the optimization differential equation or the differential inclusion, in an iteration manner, with a predetermined time step being multiplied with the value of the solved differential equation to obtain next vector variables corresponding to a next iteration time-step, until the time-step reaches the convergence time; and
outputting optimal values of the vector of variables and the cost function.
12 . The method of claim 11 , wherein the optimization differential equation is solved by a first order Euler steps:
x ( k+ 1)= x ( k )+ h.F ( k,x ( k )),
Where h>0 is the discretization time-step, and k=0, 1, 2, . . . , is the discretization index, here, F is the optimization differential equation or differential inclusion.
13 . The method of claim 11 , wherein the optimization differential equation is solved by the Runge-Kutta discretization steps.
14 . The method of claim 11 , wherein the optimization differential equation is solved by a disctization steps:
15 . The method of claim 11 , wherein the optimization differential equation is
x
.
=
-
c
∇
f
(
x
)
p
[
∇
2
f
(
x
)
]
r
∇
f
(
x
)
∇
f
(
x
)
T
[
∇
2
f
(
x
)
]
r
+
1
∇
f
(
x
)
,
Where the constant coefficient c,p,r, are such that c>0, p∈[1,2), and r∈R, and ƒ represents the cost function, ∇ƒ(x) represents the gradient of the cost function, and ∇ 2 ƒ(x) the Hessian of the cost function.
16 . The method of claim 11 , wherein the optimization differential equation is
x
.
=
-
c
∇
f
(
x
)
1
p
-
1
[
∇
2
f
(
x
)
]
r
sign
(
∇
f
(
x
)
)
sign
(
∇
f
(
x
)
)
T
[
∇
2
f
(
x
)
]
r
+
1
sign
(
∇
f
(
x
)
)
where the constant coefficient c,p,r, are such that c>0, p∈[1,2), and r∈R, and ƒ represents the cost function, ∇ƒ(x) represents the gradient of the cost function, ∇ 2 ƒ(x) the Hessian of the cost function, and sign(.) is the sign function.
17 . The method of claim 11 , wherein the optimization differential equation is
x
.
=
-
w
(
t
,
x
;
x
0
)
[
∇
2
f
(
t
,
x
)
]
r
∇
f
(
t
,
x
)
∇
f
(
t
,
x
)
⊤
[
∇
2
f
(
t
,
x
)
]
r
+
1
∇
f
(
t
,
x
)
,
w
(
t
,
x
;
x
0
)
=
Δ
∇
f
(
0
,
x
0
)
2
(
1
-
α
)
2
(
1
-
α
)
T
∇
f
(
t
,
x
)
2
α
+
∇
f
(
t
,
x
)
⊤
∂
∇
f
∂
t
(
t
,
x
)
.
18 . The method of claim 11 , wherein the optimization differential equation is
x
.
=
-
1
2
[
∇
2
f
(
t
,
x
)
]
r
∇
f
(
t
,
x
)
∇
f
(
t
,
x
)
T
[
∇
2
f
(
t
,
x
)
]
r
+
1
∇
f
(
t
,
x
)
(
2
l
(
t
,
x
)
∇
f
(
t
,
x
)
+
c
∇
f
(
t
,
x
)
2
α
)
,
.
19 . The method of claim 11 , wherein the gradient and Hessian are computed using numerical differentiation.
20 . The method of claim 11 , wherein the gradient and Hessian are computed using filters.Join the waitlist — get patent alerts
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